Dyck tilings were introduced by Kenyon and Wilson as a way to count pairing probabilities in the double-dimer model from statistical mechanics. They can be defined purely combinatorially as certain kinds of tilings of skew Young diagrams with ribbon tiles shaped like Dyck paths. We will show two bijections between such Dyck tilings and linear extensions of tree posets. These bijections yield two formulas (conjectured by Kenyon and Wilson) one of which enumerates Dyck tilings with a given lower path by a statistic that translates as descents in linear extensions and the other by a statistic translating into number of inversions. The bijection also leads to generalizations of certain Mahonian statistics to linear extensions. It is also intriguing that certain restrictions of Dyck tilings are in bijection with other objects called Dyck tableaux introduced related to the study of the TASEP model.